Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Physics for IIT JAM, UGC - NET, CSIR NET

Created by: Akhilesh Thakur

Physics : Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The document Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev is a part of the Physics Course Physics for IIT JAM, UGC - NET, CSIR NET.
All you need of Physics at this link: Physics

1. What is a Laurent series?

The Laurent series is a representation of a complex function f (z ) as a series. Unlike the Taylor series which expresses f (z) as a series of terms with non-negative powers of z, a Laurent series includes terms with negative powers. A consequence of this is that a Laurent series may be used in cases where a Taylor expansion is not possible.

2. Calculating the Laurent series expansion 

To calculate the Laurent series we use the standard and modi ed geometric series which are

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev (1)

Here f (z) = Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev is analytic everywhere apart from the singularity at z = 1. Above are the expansions for f in the regions inside and outside the circle of radius 1, centred on z = 0, where | z | < 1 is the region inside the circle and | z | > 1 is the region outside the circle.

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

 

2.1 Example

Determine the Laurent series for

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev      (2)

that are valid in the regions

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Solution: The region (i) is an open disk inside a circle of radius 5, centred on z = 0, and the region (ii) is an open annulus outside a circle of radius 5, centred on z = 0. To make the series expansion easier to calculate we can manipulate our f (z) into a form similar to the series expansion shown in equation (1).
So

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Now using the standard and modi ed geometric series, equation (1), we can calculate that

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Hence, for part (i) the series expansion is

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

which is a Taylor series. And for part (ii) the series expansion is

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev


2.2 Example Determine the Laurent series for

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev(3)

valid in the region Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev
Solution We know from example 2.1 that for

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRevthe series expansion is Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

It follows from this that we can calculate the series expansion of f (z ) as

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

 

2.3 Example 

For the following function f determine the Laurent series that is valid within the stated region R.

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev             (4)

Solution : The region R is an open annulus between circles of radius 1 and 3, centred on z = 1.
We want a series expansion about z = 1; to do this we make a substitution w = z 1 and look for the expansion in w where 1 < jwj < 3. In terms of w

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

To make the series expansion easier to calculate we can manipulate our f (z) into a form similar to the series expansion shown in equation (1). To do this we will split the function using partial fractions, and then manipulate each of the fractions into a form based on equation (1), so we get

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Using the the standard and modi ed geometric series, equation (1), we can calculate that

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

and

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

We require the expansion in w where 1 < |w| < 3, so we use the expansions for jwj > 1 and |w| < 3, which we can substitute back into our f (z ) in partial fraction form to get

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Substituting back in w = z 1 we get the Laurent series, valid within the region 1 < |z 1| < 3,

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

 

2.4 Example

Obtain the series expansion for

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev             (5)

valid in the region jz 2ij > 4.

Solution The region here is the open region outside a circle of radius 4, centred on z = 2i. We want a series expansion about z = 2i, to do this we make a substitution w = z 2i and look for the expansion in w where jwj > 4. In terms of w

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

To make the series expansion easier to calculate we can manipulate our f (z) into a form similar to the series expansion shown in equation (1). To do this we will manipulate the fraction into a form based on equation (1). We get

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Using the the standard and modi ed geometric series, equation (1), we can calculate that

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

We require the expansion in w where |w|> 4, so

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Substituting back in w = z 2i we get the Laurent series valid within the region |z 2i| > 4

Laurent Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

 

3 Key points

  • First check to see if you need to make a substitution for the region you are working with, a substitution is useful if the region is not centred on z = 0.
  • Then you will need to manipulate the function into a form where you can use the series expansions shown in example (1): this may involve splitting by partial fractions rst.
  • Find the series expansions for each of the fractions you have in your function within the speci ed region, then substitute these back into your function.
  • Finally, simplify the function and, if you made a substitution, change it back into the original variable.

Dynamic Test

Content Category

Related Searches

UGC - NET Physics Physics Notes | EduRev

,

Sample Paper

,

Laurent Series - Mathematical Methods of Physics

,

video lectures

,

UGC - NET Physics Physics Notes | EduRev

,

ppt

,

Free

,

Summary

,

Previous Year Questions with Solutions

,

Important questions

,

Laurent Series - Mathematical Methods of Physics

,

MCQs

,

past year papers

,

Laurent Series - Mathematical Methods of Physics

,

Extra Questions

,

Objective type Questions

,

Viva Questions

,

practice quizzes

,

Exam

,

shortcuts and tricks

,

Semester Notes

,

mock tests for examination

,

pdf

,

UGC - NET Physics Physics Notes | EduRev

,

study material

;